1. Field of the Invention
This disclosure relates generally to gravity and its uses in applications such as drilling and logging.
2. Background
A. Subsurface Gravity Background
Subsurface gravity may have began with pendulum measurements made during 1826, 1828, and 1854 by Airy (1856) who sought to determine the mean density of the earth by measuring the interval vertical gradients between the top and bottom of various mine shafts. Similar measurements were made by von Sterneck in 1883 and 1885 and by Rische from 1871 to 1902. It was not until the development of the portable gravity meter in the 1930s that extensive studies of subsurface gravity became possible. Since then, gravity measurements have been made in mine shafts to determine the densities of adjacent rocks, to determine the mean density of the earth, and to study anomalous vertical gradients caused by the large positive density contrasts associated with ore bodies. The concept of a borehole gravity meter for gravity logging of wells was first proposed in the 1950s, and development of the high-precision borehole gravity meter (BHGM) began in the 1960s.
Subsurface gravity measurements (e.g. in boreholes or in mines) are used to detect anomalous density structure or lateral density variations. These density variations are typically due primarily to stratigraphic, structural, or diagenetic effects that cause subsurface iso-gravity contours of equal gravity to depart from the horizontal. However, interpreting density variations is difficult, in part, because density models are not unique—in other words, different models can be used to describe the same density variations. In theory, an infinite number of density-volume model configurations can be devised to generate the same gravity anomaly. But, in many cases, a uniform, horizontal layered earth can be assumed as a useful model because formations surrounding gravity measurements are typically lateral or nearly-so and possess relatively-uniform densities in lateral directions. In such areas, subsurface gravity data are easily converted to accurate and unique interval density profiles.
Lateral density variations may be significant where folded strata, faults, unconformities, intrusions, or lateral variations in lithology, porosity or pore fluids (due to selective depositional or post-depositional processes) intersect or occur within detectable distances of a borehole. Under these circumstances, analysis of borehole gravity data is more difficult because equal density surfaces generally are poorly known and may be complex in shape.
While most other geophysical logging tools only sample in the local area of the borehole, the classical BHGM log samples a large volume of rock. Consequently, it is not significantly influenced by drilling mud, fluid invasion, hole rugosity, or formation damage that surrounds all wells to some degree. It is the ability of the BHGM log to yield a direct measure of in situ density and porosity and to characterize pore content that forms the basis of its application to oil and gas exploration and production.
Borehole gravity surveys have proven to be worthwhile at least because of their:                (1) High relative or absolute accuracy;        (2) Direct density response; and        (3) Ability to investigate a large volume of formation.        
Borehole gravimetry is also useful due to its ability to measure apparent density. Traditional logging tools only measure into a formation a few inches, whereas borehole gravity samples tens to hundreds of meters into the formation. One aspect of the BHGM that makes it an attractive logging tool in the petroleum industry is its ability to detect the contacts between gas, oil, and water at large distances from the borehole. It also can do this through multiple casing strings and formation damage, which decreases the effectiveness of the competing pulsed neutron tools for fluid saturation monitoring, as does low salinity water.
B. Physics of Subsurface Gravity
Gravity exploration utilizes Newton's Law of Universal Gravitation,
                    Force        =                  G          ⁢                                                                      m                  1                                ⁢                                  m                  2                                                            r                2                                      .                                              (        1        )            This law states that, between any two massive objects, there is a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Thus, the force of attraction is larger for larger masses but decreases rapidly for increasing distances between the masses. The constant of proportionality, G, is the Universal Gravitational Constant. A gravity sensor (or gravity meter) uses a very sensitive mass balance mechanism to measure the force of gravity acting on a test mass (often called a proof mass).
Using Newton's Second Law of Motion, the force on a test mass m2 can be expressed asForce=m2g,  (2)where g is the gravity or gravitational acceleration experienced by the test mass due to the force of gravity. Equations (1) and (2) can be combined to give:
                    g        =                              Force                          m              2                                =                      G            ⁢                                                            m                  1                                                  r                  2                                            .                                                          (        3        )            
Thus gravity is expressed in units of force per unit area or acceleration. In geophysics, the units of centimeters/second is called “gal”, after Galileo. Land, airborne and marine gravity surveys are typically recorded in units of milli-gal or mgal and borehole gravity surveys are typically given in units of micro-gal or μgal.
Although gravity is a tensor, most exploration gravity meters are only sensitive to the vertical component of gravity, gz, which is in the direction of the local plumbline. Therefore, the acceleration experienced by a gravity meter test mass is
                                          g            z                    =                      G            ⁢                                          m                1                                            r                2                                      ⁢            cos            ⁢                                                  ⁢            φ                          ,                            (        4        )            where φ is shown in FIG. 1.
Usually gravity exploration involves the spatial distribution of rocks whose densities and volumes are partly known or can be inferred. Because mass=density×volume, we can rewrite equation (4) as
                                          g            z                    =                      G            ⁢                                          ρ                ⁢                                                                  ⁢                V                                            r                2                                      ⁢            cos            ⁢                                                  ⁢            φ                          ,                            (        5        )            where ρ is the density and V is the volume. For practical purposes, we integrate Equation (5) over the entire volume,
                              g          z                =                  G          ⁢                                    ∫              v                                                                    ⁢                                          ρ                                  r                  2                                            ⁢              cos              ⁢                                                          ⁢              φ              ⁢                                                ⅆ                  V                                .                                                                        (        6        )            Equation (6) calculates the vertical component of gravity at any point due to any mass, where mass is defined in terms of density and volume. Therefore density models can be constructed using Equation (6) to simulate proposed geologic models and their gravity fields so that the models can be compared with actual measured gravity values. There are many commercially-available software packages available for doing this. Many of them are based on an algorithm developed by Talwani (1965).
If the earth is assumed to be non-rotating and perfectly spherical with a radially symmetric density distribution, with a radius of R and a mass of M, then, from Equation (3), the gravity on the earth's surface will be
                    g        =                  G          ⁢                                    M                              R                2                                      .                                              (        7        )            and the free-air gradient of gravity at the surface of the earth will be
                                          ∂            g                                ∂                          r                              r                =                R                                                    =                              -                          8              3                                ⁢          π          ⁢                                          ⁢          G          ⁢                                          ⁢                                    ρ              _                        R                                              (        8        )            where {overscore (ρ)}R is the bulk mean density of the earth. At any point inside the earth at a radius of r from the earth's center, the mass of a sphere of radius r is given by
                              m          r                =                  4          ⁢                                          ⁢          π          ⁢                                    ∫              0              r                        ⁢                                          ρ                ⁡                                  (                  r                  )                                            ⁢                              r                2                            ⁢                                                          ⁢                              ⅆ                r                                                                        (        9        )            where ρr is the internal density as a function of r. Substituting mr in Equation (9) for M in Equation (7) gives the gravity at this point (Benfield, 1937)
                    g        =                                            4              ⁢              π              ⁢                                                          ⁢              G                                      r              2                                ⁢                                    ∫              0              r                        ⁢                                          ρ                ⁡                                  (                  r                  )                                            ⁢                              r                2                            ⁢                              ⅆ                r                                                                        (        10        )            because the net attraction of a spherical shell between r and R is zero (Ramsey, 1940). The vertical gradient of gravity at this point is
                                                        ∂              g                                      ∂              r                                =                                                                      4                  ⁢                  π                  ⁢                                                                          ⁢                  G                                                  r                  2                                            ⁢                                                ∂                                                                                                          ∂                  r                                            ⁢                                                ∫                  0                  r                                ⁢                                                      ρ                    ⁡                                          (                      r                      )                                                        ⁢                                      r                    2                                    ⁢                                      ⅆ                    r                                                                        +                                          ∫                0                r                            ⁢                                                                    (                    r                    )                                    2                                ⁢                                  ⅆ                  r                                ⁢                                                      ∂                                                                                                                      ∂                    r                                                  ⁢                                                      4                    ⁢                    π                    ⁢                                                                                  ⁢                    G                                                        r                    2                                                                                      ,                            (        11        )            which reduces to
                                                        ∂              g                                      ∂              r                                =                                    4              ⁢              π              ⁢                                                          ⁢              G              ⁢                                                          ⁢                              ρ                _                                      -                                          8                3                            ⁢              π              ⁢                                                          ⁢              G              ⁢                                                          ⁢                                                ρ                  _                                r                                                    ,                            (        12        )            where {overscore (ρ)} is the density of an infinitesimally thin spherical shell of radius r and {overscore (ρ)}r is the mean density of the interior sphere of radius r. See Airy (1856), Miller and Innes (1953), Gutenberg (1959), Hammer (1963), and Beyer (1971) for further discussions. From Equation (8) we see that the second term in Equation (12) is the free-air vertical gradient of gravity for a non-rotating spherical earth.
In order to predict the gravitational field of the earth precisely at any point on the earth, we must know and correct for the shape and density distribution with the greatest possible accuracy. The earth is not actually a sphere because it rotates and thus bulges at the equator and flattens at the poles. Its shape can be closely approximated by an oblate spheroid with an eccentricity of 1/297. The rotation and general ellipsoidal shape of the earth can be taken into account by replacing the second term in Equation (12) with the normal free-air vertical gradient, which can be determined by using the spheroid model based on a best-fit reference model of gravity at mean sea level as a function of latitude φ. This reference standard model is established by the International Union of Geodesy and Geophysics. The accepted model was last updated in 1967 and isg1967=[9.7803090+0.058552 sin2 φ−5.70×10−5 sin2 2φ] m s−2  (13)or in ft s−2g1967=[32.0875312+0.192099 sin2 φ−1.87×10−4 sin2 2φ] ft s−2.  (14)
This reference standard model says that gravity varies at mean sea level from 978,030.90 mgal at the equator to 983,884.10 gal at the poles, which is a total range of 5,853.20 mgal. Note that the gravity is larger at the poles than at the equator because the equator is farther from the earth's center of mass than are the poles. The normal free-air vertical gradient is
                    F        =                                            ∂              γ                                      ∂              h                                =                      0.094112            -                          0.000134              ⁢                                                          ⁢                              sin                2                            ⁢              ϕ                        -                          0.134              ×                              10                                  -                  7                                            ⁢              h                                                          (        15        )            where h is the elevation in feet. The normal free-air gradient of gravity varies from the equator to the poles by less than 0.2% and with elevation by about 0.01% per 1,000 feet or 0.05% per kilometer (see Hammer, 1970). These variations are very small for gravity surveys made in boreholes and can be approximated by
                    F        =                                            ∂              γ                                      ∂              h                                =                      0.09406            ⁢                                                  ⁢            mGal            ⁢                          /                        ⁢                          ft              .                                                          (        16        )            
When this is substituted for the second term in Equation (12), we obtain the vertical density {overscore (ρ)} of the laterally adjacent rocks for the case in which the earth possesses a radially symmetrical distribution of density
                                          ∂            g                                ∂            r                          =                                            4              ⁢              π              ⁢                                                          ⁢              G              ⁢                                                          ⁢                              ρ                _                                      -            F                    =                                    4              ⁢              π              ⁢                                                          ⁢              G              ⁢                                                          ⁢                              ρ                _                                      -            0.9406                                              (        17        )            or, changing from elevation, h, to depth, z, we haveΔg=FΔz−4πG{overscore (ρ)}Δz  (18)orΔg=0.09406Δz−4πG{overscore (ρ)}Δz  (19)
Equation (18) is a fundamental equation of borehole gravity. The 4πG{overscore (ρ)} term applies to an infinitely extended horizontal layer of thickness Δz. According to Equation (18), gravity increases downward at a rate determined by the difference between the free-air vertical gradient F, which is essentially a constant, and a gradient of opposite sign, 4πG{overscore (ρ)}, that varies as the density of the adjacent rocks change. The positive term is due to an increase in gravity downward caused by closer approach to the center of the earth and the negative term is twice the attraction of an infinitely extended horizontal layer of thickness Δz. Therefore, increases in layer density correspond to decreases in the interval vertical gradient Δg/Δz, and vise versa. Thus it is often possible to accurately determine very small variations in the density of rocks bracketed by different Δz intervals with little or no analysis beyond the simple reduction of the basic gravity and depth measurements, which is one of the great strengths of borehole gravity.
C. Present Borehole Gravity Logging Technology
FIG. 2 illustrates a conventional borehole gravity survey. At present, borehole gravity logging is conducted using a BHGM, which consists of a single gravity sensor located in a logging sonde (10). The BHGM logging sonde is lowered down the well (12) on the end of a wireline (14). All the measurements are taken at stationary locations because any gravity sensor acceleration will be measured in addition to the gravity (due to the Principle of Equivalence). Measurements are taken at (16) and (18) in order to determine the apparent density of the intervening layer. The first measurement is taken at (16). The gravity measured at (16) is due to the mass of the intervening slab of density ρ and of thickness Δz, as given by Equation (18), plus the gravity due to the mass of the rest of the universe,g(10)=2πGρΔz+g(universe).  (20)The second measurement is taken at (18). Similarly, the gravity measured at (18) is due to the mass of the intervening slab of density ρ and of thickness Δz plus the gravity due to the mass of the rest of the universe,g(12)=−2πGρΔz+g(universe)  (21)The difference between these two measurements isΔg=g(12)−g(10)  (22)Δg=−4πGρΔz,  (23)which gives
                    ρ        =                                            -              1                                      4              ⁢              π              ⁢                                                          ⁢              G                                ⁢                                                    Δ                ⁢                                                                  ⁢                g                                            Δ                ⁢                                                                  ⁢                z                                      .                                              (        24        )            Therefore, the density of a layer of rock intersected by a well can be directly determined by measuring the gravity on either side of that layer and the distance between those gravity measurement points.
Equations (18) and (24) are not strictly correct when applied to the real earth. Departures of the earth's surface from an ellipsoid and lateral density variations in the subsurface contribute to the variation of gravity in the subsurface. Unwanted or extraneous accelerations caused by topography or mass disturbances connected with the well are usually negligibly small and can be ignored or corrected-for with sufficient accuracy. The corrections for terrain can be found in most general geophysical textbooks (e.g. Dobrin, 1976; Telford et al., 1976) and will not be discussed here.
Present commercially-available borehole gravity meter instruments are only capable of making measurements to a deviation of less than 14 degrees from vertical. Some prototype borehole gravity meters have been developed that can make measurements up to a deviations of as much as 115 degrees from vertical. FIG. 3 is an illustration of such a borehole gravity meter (20). In principle, the gravity sensor (22) is of any design that can make gravity measurements sufficient to meet its applications and that can be placed in a small enough housing to accommodate its use in a borehole logging environment—see for example Ander et al., 1999b. In practice, such sensors primarily consist of either a metal or a fused quartz relative spring sensor element. The gravity sensor (22) housing is gimbaled about its horizontal axis (24) and its vertical axis (26). The gravity sensor (22) is oriented to vertical by activating motors (30A) and (30B) that are connected to the sensor axes (24) and (26) through a series of gears (28A), (28B), (28C) and (28D). The motors are controlled by commercially-available high precision levels usually located in the gravity sensor housing.
Once borehole gravity instruments are capable of routinely making measurements at significant deviations, then corrections for well deviation can become important. Corrections for borehole deviation is generally straight forward. The correction isΔztrue=Δzmeasured cos θ  (25)It converts the measured depth to true vertical depth, where θ is the deviation from vertical. Rivero (1971) and Fitchard (1981) have developed more sophisticated corrections for the case where the well deviation involves doglegs. If corrections are not made for deviations, then the interval gravity gradient can be erroneously low, calculated interval densities can be erroneously high, and interval porosities can be erroneously low.
If a well is significantly deviated, then the theoretical latitude-dependent horizontal gradient of total gravity and any anomalous horizontal gradients of total gravity will contribute to the gravity variations measured down the well. The theoretical latitude-dependent gradient does not exceed 0.2 μgal/ft and applies only to the north-south component of deviation. The actual value is given by 0.8122 sin 2φ mgal/km, where φ is latitude (Nettleton, 1976, p. 80–81). Anomalous horizontal gradients in total gravity occasionally are greater than 1.9 to 3.8 μgal/ft. Values of the theoretical latitude-dependent horizontal gravity gradient together with estimates of anomalous horizontal gradients taken from surface gravity maps can be analyzed with hole azimuth and hole angle data from the well directional survey to determine if corrections for horizontal gravity gradients are necessary. Using present borehole gravity meter technology, in the vast majority of cases, corrections for these effects are unnecessary or are very small.
Once borehole gravity measurements can be made in horizontal or near horizontal wells, then the use of Equation (24) breaks down because Δz→0 and there is no longer a significant vertical gradient between successive gravity measurements. In the horizontal well logging case, a gravity logging survey can be treated like a horizontal land gravity profile.
D. Shortcomings
Despite the technology outlined above, significant shortcomings remain relating to gravity measurement and the use of gravity measurements in applications such as logging and drilling. Namely, conventional technology does not provide for the ability to (a) perform gravity well logging using arrays of gravity sensors; (2) create density pseudosections; (3) make gravity measurements while drilling; and (4) steer a drill bit or other apparatus using gravity measurements. Techniques of this disclosure, however, address these shortcomings, as discussed in detail below.